Calculating-machine



UNITED STATES PATENT OFFICE.

J. W. NYSTROM, OF PHILADELPHIA, PENNSYLVANIA.

CALCULATINe-MACHINE.

Specification. of Letters Patent No. 7,961, dated March 4, 1851.

To all whom 25 may concern:

Be it known that I, JOHN VILLIAM NYS- 'rRoM, of the city and county ofPhiladelphia and State o-f Pennsylvania, have invented a new and usefulCalculating-B/Iaeliine, of which the following is a full, clear, andexact description, reference being had to the accompanying drawing,which forms part of this specication, and in which- Figure l is a viewin perspective of one of my machines. Fig. 2 is a vertical sectionthrough the center of the same, and Fig. 3 is a plan of a portion of thedisk showing the method of laying out the curved lines.

lVIy invention is based upon the fact that the multiplication anddivision of numbers may be performed by the addition and sub'- tractionof their logarithme, and my machine is constructed in such mannerthatthis addition and subtraction is effected by moving a pair ofgraduated radial arms upon a disk on whose surface a series of curvesare drawn, which by their intersection with the arms show the value ofthe result. The curves on the disk in combination with the radial armsserve, not only to effect ordinary arithmetical calculations but also tosolve trigonometric formulae.

The'machine, as represented in the accompanying drawing, consists of adisk A of metal or some other suitablematerial which is mounted uponthree feet a, a,

An upright spindle B is secured at its center on which the hubs of tworadial arms C, D, are constructed to turn. These arms extend to theperiphery of the plate, their outer extremities being fitted with clampscrews Z) b by means of which they can be made fast in any requiredposition. The hub of one arm is fitted to a sleeve c o n which the hubof the other turns freely; the latter is surmounted by a clamp nut Ewhich is screwed upon the sleeve c so that by tightening or slacking itthe hubs of the two arms can be clamped to each other or can be leftfree to turn independent-ly. As

the divisions on the radial edges of the armsv nals, the otherlogarithmic curves.

In order to draw the lines of the outer scale, commence by drawing lthetwo concentric circles F, G, which bound the scale; divide the innercircle G into a number of equal parts, which number mfst be a multipleof ten; and each ten 'of which constitute a complete scale, the othersets of ten being exact duplicates of the first. In the instrumentrepresented in the accompanying drawings the circle Gr is divided intotwenty equal parts (ILOXQ) which are numbered 0, l, 2, 3,-9; 0, 1,' 2,3,-9; through the divisions 1,-2, &c., of the circle draw the radiallines e e v&c., and subdivide the arc of the circle between the radiallines e c into ten equal parts, through which draw the secondary radiallines f, f2, &c. Divide any one of the radial lines (f) into ten equalparts and through the divisic-ns draw the concentric circles, g', g2,&c. Unite the points of intersection of the radial lines f and circles gby a curved line 7L which line will be the diagonal line of the scale.All the diagonals of the scale are alike and when one is obtained theothers can be drawn from it.

That portion of the radial edge 2', j of each arm C, D, which iscontained between the two concentric circles F, G, is divided into tenequal parts, which are numbered (0, l, 2, 8 9) both from within outwardand from without inward; each of these larger divisions is subdividedinto ten equal parts, and if greater nicety is required in determiningthe results, each of these subdivisions may be divided into tenths. Thelogarithmic curves 7c o-f the outer scale are drawn from without inward,the first curve 7c starts from that point Z in the outer circle where itis intersected by the radius which passes through the 0 point on theinner circle, Gr. In order to obtain the other points of the curverecourse must be. had to a table of logarithms; commence with thelogarithm of 11. rFhis is .0414, move the radial arm C from the 0 pointof the scale in the direction indicated by the arrow m until thediagonal 7L intersects that division on the arm which count-ing fromwithin outward denotes the number .0414, or 0 of the large divisions onthe inner circle G, 4 of the large divisions on the arm C and lt oft-lie subdivisions. l/Vhen the radial arm is at this position the pointwhere its radial edge intersects t-he inner circle G is the stoppingpoint of the iirst logarithmic curve, While the point where the radialedge intersects the outer circle F is the division 1 or the startingpoint of the second logarithmic curve.

The intermediate points of the first curve are found by taking thelogarithms of 10.1, 10.2, 10S-10.9, and setting the arm in succession inthe positions in which the diagonal line 7i intersects the divisions onthe radial edge which correspond with the logarithme of these numbers;those points in the concentric circles g-g", where the radial edge ofthe arm successively crosses them will be the intermediate pointsthrough which the logarithmic curve is drawn. The stopping point of thesecond logarithmic curve 702 is formed by moving the arm C in thedirection of the arrow until the diagonal line corresponds with thatdivision on the arm which denotes the logarithm of 12; and theintermediate points are found in the same manner as those of the firstcurve by taking the logarithms of the numbers 11.1, 11.2, 11.3, &c. Theother logarithmic curves are laid out in the same manner as those abovedescribed and if great accuracy is desired a greater number than 10 ofthe intermediate points should be determined.

From the above description it will be pei'- ceived that the numbers onthe inner circle and on the arm counting outward, denote the logarithmsof the corresponding numbers on the outer circle and arm countinginward. Thus if the arm be set at the small division 2 of the outercircle which denotes the number lg or 12, the number .0792, where thediagonal line L crosses the edge of the arm, is the logarithm of 12.

The inner scale contains but one variety of curves, these are laid outlin a manner similar to that described in laying out the curves of theouter scale. As angles are measured by the degrees of the arc of thecircle subtended by their radii, and as these degrees are subdividedinto sixty equal parts, generally called minutes, it is necessary todivide those portions n, o of the radial edges of the arms C, D, whichare contained between the bounding circles H, I of the scale, into sixtyequal parts, each of which denotes of a degree or one minute and if thedimensions of the scale are sufficient these minute divisions should besubdivided into fractional parts. As the disk of the instrumentrepresented in the accompanying drawing is of small size, it wasconvenient to construct the inner scale single, instead of double as wasthe case with the outer scale.

The curves of the inner scale are laid out by the aid of the outer scaleand the graduated arms. As the outer scale is double, one semicircularportion of 10 divisions may be taken for units while the divisions ofthe other portion will represent the decimal fractional parts of a unit.The curves 7) are for the purpose of finding tlie sines and cosines ofangles, from which the other trigonometric functions can easily bedetermined. ln order to find the starting points of the curves recoursemust be had to a table of natural sines; commence with tlie sine of onedegree (1) this is .01745, move the arm C from the O point of thatportion of the outer scale which is to be used for the fractional partsof a unit, in the direction indicated by the arrow m, until thedivisions 1 and {T on the outer circle F are passed and the division4f?, on the arm (counting from without inward) intersects thelogarithmic curve, leading from the division {[6; then that point in thecircle H which is cut by the radial edge of the arm is the division 1whence the trigonometric curve 7; is drawn; and that point where theradial edge of the arm crosses the inner circle l is the division 89where the curve p stops by means of which the sines of the fractionalparts of the angle of 1o are determined. In order to find the division2, take from the table the natural sine of two degrees (2O) .03480, andmove the arm in the direction of the arrow m which l shall henceforthcall forward) until the division 3% on the outer circle F is passed andthe division (8) on the arm, intersects the appropriate logarithmiccurve; then the point where the radial edge of the arm crosses the innercircle I is the division (S8) where the trigonometric line 2) stops;while the point where the radial edge of the arm crosses the outercircle H is the division (2) whence the second trigonometric line p2starts. The corresponding starting and stopping points for the othertrigonometric lines 3, 4, 5, &c. are determined in the same manner bytaking the natural-sines of 2O, 30, 4 &c. The intermediate points ofeach trigonometric curve, are formed by drawing a series of concentriccircles r ft2-p, and by taking from a table the natural sines of thoseminutes which fall between each degree and the next succeeding one; thusfor the line 79, if nine concentric circles be used, the natural sinesof 10.6, 10.12', 10.18, 10.24, 1.80,-1.54 must be taken and the arm C isset in succession to those positions on the outer scale which indicatenumbers corresponding with the natural sines of these angles. Then thepoints s, s2, s3, st, &c. where the arm successively crosses theconcentric circles r will be the intermediate points through which thecurve p is to be drawn. The intermediate points of the remaining curvesare found in the same manner. If great accuracy be required a greaternumber of intermediate points must be determined. As the cosine of anangle is equal to the sine of its complement the divisions on'the innercircle I and on the arm counting outward will denote the angles of whichthe numbers indicated by the corresponding divisions on the outer scaleare the cosines.

As before mentioned the principle on which my machine is based is thatthe multiplication and division of numbers can be performed by theaddition and substraction of their logarithme. Hence in calculating withthe machine it is convenient to have some place where the indexes of thelogarithms can be minuted down. This is effected by a dial on the clampnut E which is traversed by a movable hand t, which can be moved freelyfrom one division of the dial to another. The divisions are numbered inopposite directions from a 0 point, the -ldivisions indicating that thenumber calculated is greater than unity, and thedivisions indicatingthat it is less than unit-y.

In the above description I have supposed that the various lines on thedisk are drawn by hand but in manufacturing calculating instruments forsale I propose to construct machines for drawing the lines by the aid ofmechanical devices.

The instrument thus constructed can be used for the additionV andsubtraction of numbers. Zhen used for this purpose the one arm C isclamped by the screw b at the O point of the circle G. The other arm Dis then moved forward until the divisions on the inner circle G and onthe arm D (counting outward) which are cut by the appropriate diagonalline correspond with one of the numbers to be added. The two arms arenow clamped together by turning the clamp nut E, the clamp screw of thearm C is slackcned and the two arms clamped t-ogether are moved forwarduntil the divisions on the circle G and on the arm C (counting outward)which correspond with the sec ond number to be added are cut by theappropriate diagonal line; then the division on the arm D cut by theappropriate diagonal line, and on the inner circle G show the sum of thetwo numbers. Thus suppose that 25 is to be added to 32; set the arm C atO, the arm D at 25, clamp the two move the two so clamped past thedivision 3 on the circle G until the arm C is in such a position thatthe diagonal line cuts the division 2 (counting outward). Then thedivision 7 on the -priate logarithmic curve.

ample, is to be added to the sum thus found clamp the arm D in itsposition, slacken the nut E and move the arm C back to 0. Clamp the twoarms together, `unclamp the arm D, and move the two until the arm Carrives at the division 2 on the inner circle, then the division (7 onthe arm D cut by the diagonal line, preceded by the number of thedivision (7 on the inner circle last passed by the arm D show the sumsought which is 77..

It will be perceived from the above that the operation of addition iseffected by increasing the angle included between the two arms inarithmetic proportion to the sums to be added: Subtraction is thereverse of addition hence it is effected by diminishing this angle inarithmetic proportion. Thus for example let us suppose that 2O is to besubtracted from 97. Set the arm D at 97, and the arm C at 20; clamp thetwo, and move them so clamped backward (that is in` adirection thereverse of the arrow) until the arm C arrives at the O point. Then thedivision (7) on the arm D cut by the diagonal line preceded by the nextdivision (7) on the inner circle shows the remainder, 77, sought.

vWhen multiplication is to be performed it is effected by increasing theangle between the two arms in proportion to the sums of the logarithmsof the numbers. Thus, to multiply 245 by 122, move the arm D forwardpast the division 2%@ on the outer circle F until the division 5 on thearm (counting inward) is cut by the logarithmic curve, and clamp it inthis position; as the logarithmic index of 245 is 2 the hand t.

must be moved to the division -l- 2 on the dial, add to this the index 2of the multiplier (122) making -let. Place the other arm C at thedivision 1 on the outer circle, clamp the two arms together and movethem so clamped forward until the arm C has passed the divisions lZ ofthe outer circle, and the appropriate logarithmic curve cuts thedivision 2 on the arm. It will now be found that the division on the armD cut by the logarithmic curve is Sf, which preceded by the divisions(29) on the circle F passed by the arm, gives the iig. 2989 of thcproducts, and as the index is -l- 4c the number is 29890. If thisproduct is to be multiplied by a second multiplier, the arm D is clampedat its position and the arm C is moved back to the starting point (l) ofthe scale; the two are now clamped and moved in connection until theproper division of the arm C is cut by the appro- The division of thearm D then cut by the appropriate logarithmic curve,'preceded by thefigures of the divisions of the circle F passed over show the figures ofthe new product. Thenever in moving the clamped arms the forward onepasses the dividing point of the double scale, the logarithmic index isincreased by 1, and hence this increase must be minuted by moving thehand t on the dial.

As division is the reserve of multiplication, it is found by diminishingthe angle between the two arms. The arm D is set at the division of thescale corresponding with the dividend, and the arm C is set at that ofthe divisor; the two are then clamped and moved backward until the arm Creaches the starting point of the scale when the divisions of the arm Dpreceded by the figures on the outer circle, will show the quotient.

If the logarithm of a number is required it is found by setting one ofthe arms to the division on the outer circle and arm which correspondswith the number. The division on the arm counting outward cut by thediagonal line and preceded by the figures of the divisions on the circleGr passed by the arm in moving from the starting point is the logarithmrequired.

If a root is to be extracted, as for example the cube root of 27, setone of the arms at that number on the outer circle F. The correspondingnumber (31) on the arm counting outward cut by the diagonal line andpreceded by the number of the division (el) last passed by the arm willthen be the logarithmic number of 27 or 431, the logarit-hmic index of27 is 1 and the whole logarithm is consequently 1.431; divide this bythe exponent of the root 3, making 0.477, and set one of the arms tothis logarithm. Then the corresponding divisions 800 on the outer circleand arm are the figures of the root sought, and as its index is O, thenumber is 3. Powers, as squares, cubes, &c., are found by reversing`this operation or by first iinding the logarithm of the num` ber,multiplying it by the expo-nent and setting one of the arms to the newlogarithm, when the number indicated by the outer circle and armcounting inward is the power sought.

The inner scale is used to find the sines and cosines of angles, andwhen these are Vfound to determine from them the other trigonometricfunctions by the usual formulae. lt is also used to multiply or dividetrigonometric functions, in which case the operation is effected bymoving the arms in the same manner as when multiplying or dividing`numbers on the outer scale. The

result of the operation is indicated by the divisions on the outercircle of the outer scale and the divisions of the arm counting inward,and the logarithm of the result is let it be required to solve theequation sin O o sin. B I T the values of c B and Z) being as follows:@2170+ B225o :112-{; set the arm C at the start-ing point and move thearm D forward to the position corresponding with the number 17.0; clampthe two arms together; set the finger t to the proper index +2. Now movethe two arms forward until the arm C arrives at the positioncorresponding with the sine of 25O as shown by the divisions on theouter circle H of the inner scale, clamp the arm D, slacken the clampnutE and set the arm C to the position corre-` sponding' with'the number112; reclamp the two arms and as the logarithmic index of 112 is -l-2this must be subtracted from the number pointed at by the linger on thedial leaving 0 for the index of the result. The two arms now clampedmust be moved backward until the arm C arrives at the starting pointwhen the number of the divisions on the inner scale indicated by the armD is the angle C required or 390, 58.

The above examples are sufficient to enable persons skilled inarithmetical and trigonometrical calculations to use my machine and Itherefore deem it unnecessary to give others.

In order to facilitate the working of the machine I have litted a springstop U to the disk at the starting point of each scale, so that insetting the arm at this starting point all that is necessary is to bringit in contact with the stop.

I have thus far described the logarithmic curves as curving forward asthey extend. from the outer circle toward the center of the scale, whilethe diagonals curve forward as they extend from the inner circleoutward. This arrangement of the two curves is merely one of conveniencefor it is evident that the same results can be obtained by reversing thedirections o-f drawing the two curves; that is to say by making thelogarithmic curves start from the inner circle and the diagonals fromthe outer one. The figures on the inner divisions would then denote thenumbers while those on the outer circle would denote their correspondinglcgarithms. lt is also evident that instead of laying out the curves ofthe inner scale by means of a table of natural sines and the outerdivisions of the outer scale, recourse may be had to a table oflogarithmic sines, in which case the arm must be set to its place by theinner divisions of the outer scale which denote the logarithmic numberscorresponding with the natural numbers of the outer divisions.

`What I claim as my invention and desire to secure by Letters Patentisl. The logarithmic curves of the outer scale in combination with thediagonals and graduated arms, the curves being laid out Substantially inthe manner herein set forth.

2. I claim the trigonometric curves of the inner scale in combinationWith the gradu ated arms and logarithmic curves of the outer scale, thecurves being laid out substantially in the manner herein described.

3. I claim the two graduated arms constructed in such manner that theycan be moved in connection or independently substantially in the mannerand for the purposes herein set forth.

In testimony whereof I have hereunto 20 subscribed my name.

J. W. NYSTROM.

Witnesses:

P. A. WATSON, E. S. RENNICK.

